Suppose the map $f:M\rightarrow N$ is differentiable, where $M,N$ are $m$-manifolds with $M$ compact and $N$ connected. If the degree of $f$ is 1, then $f$ is surjective?

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    $\begingroup$ A non-surjective map has degree $0$, because if $x\in N\setminus f(M)$, then $x$ is a regular value of $f$. $\endgroup$ – user228113 May 9 '16 at 23:06

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